A Uniform Display System For Intuitionistic And Dual Intuitionistic Logic

We give a way to \display" propositional intuitionistic and dual-intuitionistic logic in one uniform setting. We show how the usual \singletons on the right" restriction of Gentzen's LJ, and the dual \singletons on the left" restriction can be mimicked in Display Logic. The resulting Display System JDJ captures a hybrid logic that permits reasoning with partial and paraconsistent knowledge since it contains two negations : and with the following properties: A _ :A is not a theorem; A^ A is not a contradiction; A ^ :A is a contradiction; and A_ A is a theorem. We then show how to obtain a Display system for Classical Logic by the addition of structural rules only.